Non-Ideal Solutions

Henry's Law

Many solutions are not ideal. For ideal solutions the role of intermolecular interactions can be ignored. This may be because they are small or because two components have the same interaction with each other that they have with themselves. In other words similar solvents will form ideal solutions. However, in many cases, intermolecular interactions cause deviations from Raoult's law. We can consider the "like" interactions between molecules of same species and "unlike" interactions between molecules of different species. If the unlike-molecule interactions are more attractive than the like molecule interactions, the vapor pressure above a solution will be smaller than we would calculate using Raoult's law. If the unlike-molecule interactions are more repulsive, then the vapor pressure is greater than for the ideal solution. As shown on Page 978 of M&S attractive interactions between unlike molecules leads to negative deviations from Raoult's law (lower vapor pressure than ideal) and repulsive interactions lead to positive deviations (higher vapor pressure than ideal).

As any solution approaches a mole fraction of one (i.e. approaches a pure solution of one component) it becomes an ideal solution. In other words, P₁ à x₁P₁* as x₁ à 1. However, as x₁ à 0 the component is surrounded by unlike molecules and the solution has the maximum deviation from ideal behavior. For this case we define Henry's law, P₁ à x₁k_H,1 as x₁ à 0. In this expression k_H,1 is the Henry's law constant. Although we have focused on component 1 the same holds true for component 2.

A general type of expression for non-ideal behavior is shown below.

Note that the mole fraction of component 2 appears in the exponent. A plot of this function for a = 1 and b = 1 (blue) compared to Raoult's law (black) is shown in Figure below.

For the example shown in the plot above we have assumed that P₁* = 100 torr. Note that as x₁ à 1 the slope approaches the ideal slope obtained from Raoult's law. However, as x₁ à 0 (and therefore x₂ à 1) the slope is quite different from the ideal behavior. We can see that as x₂ à 1 the slope becomes P₁*e^a+b. Note that this is value of the Henry's law constant k_H,1. This is depicted in the Figure below.

You can see from the Figure that the Henry's law value can be quite different from the ideal value given by Raoult's law. The value of the Henry's law constant is given by the slope of the dotted purple line.

The really powerful aspect of this analysis comes from the Gibbs-Duhem equation. In Gibbs-Duhem we find the relationship between the vapor pressure for compound 1 and compound 2 in a solution.

Let's consider the general case:

Gibbs-Duhem permits calculation of g and d in terms of a and b for component 1. Let's see how this works. From Gibbs-Duhem

The last step arises because x₁ = 1 - x₂ and therefore dx₁ = -dx₂. A similar expression holds for m₁.

In the vapor we have

To obtain dm₁ here we take the derivative with respect to x₁.

Thus,

Now, change variables to x2

Integrate from x2 = 1 to arbitrary x2.

Using the fact that

We obtain

or

Thus,

If assume that a second component has a vapor pressure of P₂* = 200 torr and then use the above information (e.g. a = 1, b = 1 so g = 5/2 and d = -1) then we obtain the following plot.

The second component is shown in the dotted black (Raoult's law) and blue lines. For the second component the Henry's law constant is k_H,2 = P₂*exp(3/2) = 896 torr. To obtain this value we substitute in P₂* = 200 torr and x₁ = 1. As stated above, the value of the Henry's law constant is equal to the slope of the line that is tangent to the curve x₁ = 1.

The deviations shown here are all positive deviations from Raoult's law. It is clear that positive deviations in one component will lead to positive deviations in the second.

Azeotropes

The phase diagram for an azeotrope shows a minimum boiling point at some composition other than the pure substance. For example, ethanol and benzene form an azeotrope. The phase diagram shown in Figure 24.11 of M&S indicates that the boiling point of a solution of x_benzene = 0.4 is around 66 ^oC whereas pure benzene boils at 80 ^oC and pure ethanol boils at 78.5 ^oC. The name azeotrope refers to the fact that the composition of the liquid does not change once this composition is reached. Moreover, if we start out at some other mole fraction we will eventually arrive at the azeotropic mole fraction since it has the lowest boiling point. It is not possible to separate the liquids by distillation in this case.

Activity

The activity in non-ideal solutions corresponds to mole fraction in ideal solutions.

The activity replaces mole fraction in the expression for the chemical potential

When considering a non-ideal solution the above expressions hold and thus the mole fraction x_j is no longer equal to P_j/P_j*. However, as the mole fraction approaches unity (a pure substance) the solution becomes ideal. Thus, as x_j à 1, a_j à x_j. The ratio of the activity to the mole fraction is called the activity coefficient g_j.

So as a solution becomes ideal g_j à 1, as well.

If we examine the expressions above for the Henry's law behavior of a solvent we see that the exponential terms that were used are related to the activity coefficient. Starting with the example from above:

we see that

and

The entire discussion above regarding the use of the Gibbs-Duhem equation shows that activities of one component in a mixture can be calculated from the activity of the second component.

See example 24-8 in M&S for a simpler example.

Definition of Standard State

The definitions of activity or chemical potential are only meaningful relative to a standard state. There are two possible standard states. These are the Raoult's law standard state and Henry's law standard state. The Raoult's law standard state applies when two solvents are completely miscible in all proportions. The Henry's law standard state applies when one component is sparingly soluble in the other. The Raoult's law standard state is the same as the standard state in an ideal solution as the mole fraction approaches one.

However, if component j is sparingly soluble Henry's law P_j à x_jk_H,j as x_j à 0. Thus, the chemical potential is

The activity of component j is defined by

In this case we define a_j by

And our reference state is

The numerical value of the activity and the chemical potential depends on which definition we use. This is discussed in an example in M&S on pages 991-992.

The Free Energy of Mixing

The free energy for formation of a solution from individual components is given by

Since

G^soln = n₁m₁ + n₂m₂ , G₁* = n₁m₁* and G₂* = n₂m₂*.

We have

For an ideal solution there is not enthalpy of mixing. The volume of the mixture also does not change for an ideal solution. The entropy change can be obtained from

in agreement with a previous derivation. For non-ideal solutions we use the definition of the chemical potential from above to obtain

We can break DG_mix into two parts conceptually.

The ideal free energy of mixing is the same as the quantity defined above. The excess free energy of mixing gives the difference between the ideal and real free energy of mixing.

Regular solutions

If we consider a case where

As we saw above the activity coefficients are:

And the excess molar free energy is:

Combining this expression with the ideal free energy of mixing gives

Solutions that obey this equation are called regular solutions.